Reed-Solomon Codes Against Insertions and Deletions: Full-Length and Rate-$1/2$ Codes

TL;DR

This paper analyzes the error-correcting capabilities of Reed-Solomon codes against insertions and deletions, providing new characterizations, bounds, and a polynomial-time construction for rate-1/2 codes.

Contribution

It offers a complete characterization of 2D RS codes' limits, extends correction bounds to higher dimensions, and introduces a polynomial-time construction for rate-1/2 codes correcting single errors.

Findings

Nearly all 2D RS codes can correct up to (1 - δ)q insertions/deletions for large q.

Existence of full-length k-dimensional RS codes correcting q/(10k) errors for large q.

Polynomial-time construction of rate 1/2 RS codes correcting one insertion/deletion.

Abstract

The performance of Reed--Solomon codes (RS codes, for short) in the presence of insertion and deletion errors has attracted growing attention in recent literature. In this work, we further study this intriguing mathematical problem, focusing on two regimes. First, we study the question of how well full-length RS codes perform against insertions and deletions. For 2-dimensional RS codes, we provide a complete characterization of codes that cannot correct even a single insertion or deletion. Furthermore, we prove that for sufficiently large field size~$q$, nearly all full-length $2$-dimensional RS codes can correct up to $(1 - \delta)q$ insertion and deletion errors for any $0 < \delta < 1$. Extending beyond the 2-dimensional case, we show that for any $k \ge 2$, there exists a full-length $k$-dimensional RS code capable of correcting $q / (10k)$ insertion and deletion errors, provided $q$ is large enough. Second, we focus on rate $1/2$ RS codes that can correct a single insertion or deletion error. We present a polynomial-time algorithm that constructs such codes over fields of size $q = \Theta(k^4)$. This result matches the existential bound given in \cite{con2023reed}.